Chapter 21 - Theme and Variations

One of the charms of the simple two-dimensional dissection puzzles shown in Chapter 1 is that they construct many different simple geometrical
shapes with the same set of pieces. Some of the polyhedral block puzzles in Chapters 3 and
18 construct multiple shapes
but they are non-interlocking. The difficulty of achieving this feature with interlocking
puzzles was demonstrated in Chapter 11 by a pair of designs that
succeeded only partially. Is it possible for a set of interlocking puzzle pieces to
construct many different polyhedral shapes?

The Six-Part Invention

Recall the simple two-piece dissection of the rhombic dodecahedron shown in
Fig.
147. These half-pieces can be joined in pairs in different ways to form
puzzle pieces. Excluding those that are impossible to assemble or have an axis
of symmetry, there are 12 such pieces, shown in Fig. 175.

Fig. 175

Let us make a list of possible constructions, i.e. ways that R-D blocks can
be clustered symmetrically. To keep things simple, consider only those with six
or fewer blocks. Eight such figures are shown in Fig. 176a.

Fig. 176a

Editor's Note: Stewart named this puzzle The
Peanut Puzzle.

Fig. 176b

Now for the interesting part. Can you find a subset of six pieces from the
set of 12 that will construct all eight of the above figures? Do not be too
discouraged if not, because Beeler's computer could not either. Of the 924
possible such subsets, there is one, however, that will construct seven of the
eight figures. Find it if you can, keeping in mind that the seeking may be more
fun than the actual finding.

Considering the thousands of different possible subsets of puzzle pieces and
the many interesting constructions possible with any of them, the recreational
potential for this family of puzzles is vast and practically unexplored. Choose
your own personal subset of puzzle pieces and compile your own library of
geometrical or animated shapes that they will construct (see Fig. 177). The
pieces are also great fun just to doodle with. Any closed loop can be considered
a solution of sorts.

Fig. 177

A well-crafted set of these pieces makes a most satisfactory puzzle. Each
half of each puzzle piece is made of three squat octahedra building blocks
joined accurately together, and the full pieces are then made up of these
half-pieces joined different ways.

(Incidentally, to digress slightly, a fascinating recreation is to determine
how many of the eight constructions shown in Fig. 176 are space-filling. You may
be surprised at the answer.)

The Eight-Piece Cube Puzzle

The Six-Part Invention design leads by analogy to
one based on cubes in place of rhombic dodecahedra, dissected the same way.
These half-pieces can be joined in pairs eight different ways, as illustrated in
Fig. 178.

Fig. 178

The obvious question is whether these eight pieces can be assembled into a
cube. That they do, and much more. Here is one of the best examples in this book
of a geometrical recreation that lends itself to use in the classroom. For
example:

1.

Using two disconnected half-pieces, find all the ways that they can be
joined face-to-face. You will of course arrive at the set of eight pieces,
but this simple exercise can be quite instructive.

2.

Prove that four pieces are the fewest that can be connected together in
a closed loop. Prove that the square is the only such possible figure. Can
two separate squares be made using all eight pieces? Why not?

3.

Prove that the 2 x 4 rectangle is impossible. (Problems of this sort can
always be solved systematically by trying every piece in every possible
combination, but look for shorter and more elegant proofs using logic. Now
what other shapes cannot be made for the same reason?

4.

Assuming all solutions to be closed loops, prove that an even number of
pieces must always be used. Find all possible solutions using six pieces.
Likewise using all eight pieces. Examples are shown in Fig. 179a.

Fig. 179a

Editor's Note: Stewart named this puzzle Pieces
of Eight.

Fig. 179b

Six of the pieces have reflexive symmetry and the other two are a reflexive
pair. It necessarily follows that every solution must either be self-reflexive
or occur in reflexive pairs. (These pairs are not counted as separate
solutions.) Can you figure out why?

Some of the most fundamental questions in physics have to do with symmetry,
and perhaps this puzzle will stimulate the student's interest in this
fascinating subject. If the most elementary particles in the universe and all of
the laws governing them were symmetrical (which is not to say they are), would
it not follow that everything made from them, from atoms to the entire universe,
should be either self-reflexive or one of a possible reflexive pair? But therein
lies a curious paradox. Imagine that in the next instant the universe switched
to its mirror image. How could you tell? Would not human consciousness be
reflexive also? (Whatever that means!)

Another strange case is the DNA molecule and the genetic code. Most of us are
right-handed, nearly all of us have our appendix on the right, and all of us
carry DNA with a right-handed twist. How are instructions for right-handedness
carried genetically? Would an identical but reflected DNA molecule produce an
identical but reflected organism? A lucid discussion of these and many other
fascinating problems in symmetry may be found in the Ambidextrous
Universe by Martin Gardner, but don't expect to find all the answers.

The half-pieces for the Eight-Piece Cube Puzzle are made from three square
pyramid blocks joined together. These blocks are made from sticks of
isosceles-right-triangular cross-section with two 45-degree cuts. For
experimental work, the mating joints can be slightly on the loose side. A more
accurate model of this puzzle made of fine woods with close-fitting joints is a
delight to play with. The sharp edges may be beveled or rounded slightly to give
its stark Bauhausian functionality a little more softness and warmth.

More Variations

Although a well-crafted set of puzzle pieces for either of the two designs
just described can be quite entertaining in itself, more important for the
purpose of this book is that the geometrical principle they are both based on is
even more fun to play with. It leads along an endless trail of new discoveries.
For example, as suggested by Fig. 147, an
obvious variation of the Six-Part Invention is to use the connection with three
prongs rather than two. The three sample pieces shown in Fig. 180 assemble into
a triangular cluster.

Fig. 180

By truncating the Eight-Piece Cube pieces to convert them into cuboctahedra,
the puzzle remains the same but assumes an intriguing new geometry (Fig. 181).

Fig. 181

When any of the half-pieces described in this chapter are joined in threes
rather than pairs, the numbers of puzzle pieces, practical sets, and possible
constructions stretch the imagination. To give but one example, 12 identical
pieces assemble to form the Triple Cross Puzzle shown in Fig. 182. Could you
assemble 14 such pieces in axial symmetry? How about 46 such pieces?

Fig. 182

Now imagine combining all of the above into one super set containing singles,
doubles, and triple pieces, and perhaps some even larger. Simply as a play
construction set, who could possibly resist the urge to tinker with these pieces
and fit them together different ways? At the same time, this versatile set of
puzzle pieces contains practically unlimited potential as an educational tool
and as a kit for discovering new puzzle problems, some of which should baffle
experts. A few sample puzzle constructions are shown in Fig. 183. A large set of
such pieces, well-crafted of hardwood, makes a marvelous construction set.

Fig. 183

The Pillars of Hercules Family

When dissected cubes are combined with whole cubes, the number of puzzling
possibilities takes another quantum jump. This idea hatched just in time for
inclusion in this book, so just one simple example will be given here.

The set of seven dissimilar puzzle pieces shown in Fig. 184 could be
described as three ordinary pentacube pieces and four jointed polycube
half-pieces. They assemble one way only to form a 3 x 3 x 3 cube. The next time
someone says that 3 x 3 x 3 cube puzzles are too easy, give them this one
disassembled. Note that the half-pieces can be combined in pairs three different
ways, with each pairing having four variations, so right there is ordinary
puzzlement multiplied by a factor of 12!

Fig. 184

Now, if you are the type that likes to have a little extra fun sometimes,
design a 3 x 3 x 3 cubic block puzzle like the above, except having only one
jointed piece and with a tightly fitting joint. When carefully made these joints
are practically impossible to detect. With the jointed piece assembled
backwards, it looks like just an ordinary cubic block puzzle but is of course
impossible to assemble.

With millions of possible combinations from which to choose, clever designs
of this sort might incorporate many other interesting features such as
interlocking assembly and construction of multiple symmetrical problem shapes.
This leads into a whole new world of puzzling possibilities, almost totally
unexplored, hence the name Pillars of Hercules. Note also that such
puzzles are fairly easy to make in wood, since most of the blocks are just
cubes. More intriguing still is the possibility of extending this idea to
rhombic dodecahedra, with whole blocks and half-blocks combined, as suggested by
the sample pieces shown in Fig. 185. Two of them form a small tetrahedron, three
of them a triangle, and all five a large tetrahedron.